## Saturday, November 09, 2019

### All Hermitian operators are observable(s)

The main reasons behind the omnipresent denial of quantum mechanics are simply
• the ideologically rooted stubbornness that makes these people insist on the rules of classical physics
• generally absent physics-related innate aptitude
Those would-be thinkers who keep on writing completely wrong books saying "whiny whine, there is something wrong about quantum mechanics and Copenhagen, whiny whine" are simply too ideological and too dumb for modern physics. These traits define their personalities and there's no easy way to fix them: something is simply missing.

However, the universal rules of quantum mechanics don't seem too hard. They are natural, straightforward, and may be explained on several lines. An observer perspective must exist; the observer inserts the knowledge about the measured observables (Hermitian operators) in terms of the wave function or density matrix; evolves these collections of complex numbers unitarily; and predicts the probabilities of future measurements via Born's rule while every new measurement is accompanied (really: mathematically expressed) by "collapsing" the wave function into the appropriate eigenstate (the projection onto the right space of eigenstates).

Is that really so difficult? Is the barrier preventing most people from "getting" these rules intellectual or ethical? These rules are really unavoidable which seems obvious as well because
• there is clearly no truly qualitative difference between "small" and "large" objects, "small" and "large" events (measurements), so the boundary (the Heisenberg cut) must clearly be inserted by someone
• because the location of the Heisenberg cut can't be objectively well-defined, it must depend on the observer, along with all the facts
• the results obviously are random – we may see e.g. the random locations of the dots in the double slit experiments
• emulating these random numbers by pseudorandom ones would require hidden variables which would conflict with the Lorentz invariance etc.
To summarize, the outcomes of the quantum measurements are truly random; and the identification which parts of events are measurements unavoidably is subjective at least to some extent. There is no way to avoid these facts – they're vital parts of the axioms that are needed for quantum mechanics to work, to make viable predictions.

On the other hand, these facts – while "arbitrary" axioms of a proposed theoretical framework, quantum mechanics – are almost directly extracted from the experiments. We just look at the double slit experiment and we get random results. So the theory must say that the outcomes are random or pseudorandom. The pseudorandom option is excluded because the hidden variables carrying the memory for the pseudorandom results would have to be sent superluminally which would conflict with relativity.

And we do observe the phenomena to be the same in all inertial frames which means that relativity has to hold and the pseudorandom option has to be eliminated. The results are truly random. The random outcomes of the quantum experiments are the ultimate fair random generators in Nature. And they're not exceptional events: these random numbers are primary and everything else boils down to them.

Still, over the years in which I tried to explain quantum mechanics (and why its proper rules really are right) to many people – and the efforts failed in way over 90% of the cases – I wondered: Isn't there some truly technical misunderstanding that makes so many people so incredibly dense, that makes so many people write so incredibly idiotic books and popular articles?

The answer to this question mostly looks fuzzy: these people have the wrong conceptual thinking about quantum mechanics which distorts their "technical thinking" as well and vice versa – it's a vicious circle. They want to believe in classical physics which is why they try to reframe all the objects in quantum mechanics as classical objects in some way. Most typically, a wave function is a classical (observer-independent, factual) wave for them.

This incorrect classification makes them do wrong things with the wave function mathematically, too. You know, very different things are happening to classical waves e.g. in Maxwell's theory from the 19th century; and to the wave function from the mid 1920s. In particular, classical waves are evolving almost certainly in a non-linear way – Maxwell's equations are only linear in some approximations and the evolution surely isn't linear if you also include the Dirac or Higgs waves (because the truly linear evolution of all these waves would mean that there are no interactions at all).

On the other hand, the evolution of the wave function must be perfectly linear. This is absolutely needed because if there are two mutually exclusive (orthogonal) initial states, they must evolve into mutually exclusive final states. The linearity of the quantum evolution is just some complex generalization of the linearity of the evolution of probabilities (even those on the phase spaces in classical physics). This linearity of wave functions and non-linearity of the observables is an important point that makes a mathematically thinking person understand that they just can't be conflated. They are qualitatively different objects.

However, the anti-quantum zealots are generally not mathematically thinking people. They always place some kind of ideology – which they incorrectly describe as physical intuition – above the mathematical patterns. They always have the tendency to derive some physical conclusions from pure words, from the social consensus, from philosophy, and all this kind of unscientific garbage.

But I finally want to get closer to the topic announced in the title.

I think that many of the people would understand that the basic rules of quantum mechanics are unavoidable if they understood the completely technical point that for any Hermitian linear operator acting on a Hilbert space, there exists an apparatus that measures the corresponding observable. Because non-commuting observables clearly "can't be well-defined" at the same moment, it really proves that the choice of the apparatus is essential for the "chosen way of probing" the system of interest – in the sense of Bohr's complementarity.

So any physical object carries the potential to yield some result (eigenvalue) after any chosen measurement (of an operator) and these measurements can't be made simultaneously because they interfere with each other. Whenever the laws of quantum mechanics are actually verified, the choice of the apparatus – or the measured operator – is some "classical data" that is inserted to the calculation so the predictive schemes of quantum mechanics has to be given both the quantum mechanical data – the probability amplitudes – and the classical data – the Heisenberg choice informing the theory about the observable that we measure.

The simplest nontrivial example of the quantum information is one qubit, a physical object described by a two-dimensional Hilbert space. A general Hermitian operator is a $$2\times 2$$ Hermitian matrix. It carries a complex number in the upper right, the complex conjugate number in the lower left, and two real numbers on the diagonal. In total, there are "four real numbers" of information. Such a general Hermitian matrix may be written as$L = a + \vec n \cdot \vec \sigma$ where $$\vec \sigma$$ is the 3-vector of the Pauli matrices. The four real numbers mentioned above are encoded in $$a,n_x,n_y,n_z$$. Now, the effect of the $$a$$ term is trivial – it just shifts all the measured eigenvalues of $$L$$ by a simple $$c$$-number-valued additive shift, $$a\in\RR$$. And the other three parameters are conveniently written as a 3-vector.

The absolute normalization of $$\vec n$$ has a trivial effect – it just rescales the measured eigenvalues by a multiplicative factor. So the only nontrivial information about the "measured operator" is the unit vector$\hat n = \frac{\vec n}{|\vec n|}$ which determines the direction in the three-dimensional space, a point on the $$S^2$$. And we just measure the spin "up-down" along this particular direction or axis. Now, if we have an apparatus A that can measure the up-down eigenvalue of $$j_z$$ relatively to the $$z$$-axis, we may create an apparatus that measures the spin with respect to the axis $$\hat n$$ simply by rotating the apparatus A into the desired direction.

We have classified all observables in the case of the 2-dimensional space – they're equivalent to linearly rescaled up-down bit relatively to a direction on an $$S^2$$. A funny generalization is that for $$M$$ qubits, there is still an apparatus that may measure any observable. A general $$M$$-qubit Hermitian operator has $$D_M^2$$ "real parameters" in it ($$D_M=2^M$$; again, it's one-half of the complex square matrix: the diagonal entries are real). Out of these $$D_M^2$$ real parameters, $$D_M$$ determine the eigenvalues and $$D_M^2-D_M$$ determine the eigenvectors modulo the complex scaling of each.

The single qubit case is sort of trivial and looks too "discrete" or "quantum-computer-like". So it's a good idea to understand what apparatuses for more complicated observables look like. It's vital to understand how the momentum of an electron may be measured using the crystals and interference. And yes, we've implicitly switched to infinite-dimensional Hilbert spaces where $$M=\infty$$ and we need to care about the finiteness of the eigenvalues and similar things (some operators where "something important" diverges may be prohibited as observables).

But the point is that every quantum mechanical system is described by a Hilbert space and every well-defined (convergent in some sense...) Hermitian linear operator allows you to construct a corresponding apparatus (with your hands) that will measure the observable. The final stage of the measurement may be similar to what the up-down spin measurements are doing (and the Stern-Gerlach apparatus uses a magnetic field to transform the discrete internal spin information to a location). However, before one gets to this final point which "collapses" the wave function, various unitary transformations may be applied – they generalize the rotation of the $$z$$-axis to the unit vector $$\hat n$$ that I mentioned in the case of the qubit. The unitary operations of a quantum computer represent a subclass of such unitary transformations; that subclass is important in "engineering" (quantum computers) but fundamental physics itself encourages you to treat these selected unitary transformations on par with the whole continuum of $$U(N)$$ transformations.

By complementarity, Bohr meant that every quantum mechanical system is simply waiting to produce some classical information – an eigenvalue of an operator (or many such eigenvalues if many operators commute with each other) – but there are many ways how this can be done because there are many operators that don't commute with each other. This information about "how the Hilbert space is split to the relevant eigenstates which may be the states after the measurement" must be inserted for a prediction to be possible and it must be inserted by an observer.

At this point, the only "improvement" of the axioms of quantum mechanics that you could suggest is the idea that perhaps with the observer, there will be something unique and the "information about the right observables" may be uniquely calculated and doesn't have to be inserted. But this proposal is clearly wrong as well simply because an observer – typically a human – is just another bound state of electrons and nuclei, and this observer plus whatever system he or she studied may be considered a larger system that is studied by another, external observer.

For this extra, external observer, the lesson of the previous paragraph still applies: this external observer may measure any convergent operator describing the combined system ("internal observer" plus "what he measured"). And because all the observables are again "equally good" and "equally measurable" by a cleverly constructed apparatus chosen by the "external observer", it follows that there just cannot be any calculable "preferred operator" or "preferred eigenstates".

(Well, in practice, some observables describing the "internal observer" plus "his system" will be far easier to be measured than most others, due to decoherence etc. It means that in practice, there does exist some derivation of the "right observables". But this derivation is never quite precise or unique and it still depends on some input that must be specified by the external observer – who must say at least what "in practice" precisely means. The observer-dependence cannot ever be eliminated entirely, as a matter of principle.)

The information about the observable we want to measure (or we have measured) has to be inserted and this information is in principle subjective – or, more precisely, it has at least some subjective portion to it. This fact is really proven by the simple arguments above: by a combination of the fact that we may measure any observable; and by Wigner's friend setup showing that this is still true even if an observer is included as a part of the system.

This whole novel observer-dependence of the predictions of quantum mechanics may be said to mathematically boil down to the qualitative equivalence or symmetry between all observables. Whatever is observable – i.e. subject to experimental predictions in science – is described by a Hermitian operator. Because they don't commute, they can't be assumed to have "sharp values" simultaneously and before the measurement. And because there is a conceptual symmetry or equivalence between all these operators, "someone" has to break the symmetry and choose what we want to measure and predict.

This information must be inserted into quantum mechanics as an extra input. Predictions are impossible without it. And every possible way in which quantum mechanics could "insert it by itself" have been proven impossible. The source of the "symmetry breaking" between the conceptually equivalent operators is known as the "observer". The word sounds too anthropomorphic than it should; the only point is that there's some conceptual equivalency between all the operators and it has to be broken by an extra input. That's it.

There was a time when I was 15 and I didn't understand these straightforward arguments. But at some point, I did get them. The precise wording and emphasis may have depended on time but the fact that the information about "what observable is interesting for us" must be inserted is something rather basic. I have written hundreds of blog posts that try to convey the same simple point – and its proof – in many different ways, using various words. All these blog posts are extremely talkative because I think that an intelligent person "gets it" after observations that could be shorter by a factor of ten than this single blog post.

Yet, even long blog posts and hundreds of them don't seem to be enough. Too many people are just too stupid and/or too ideologically blinded to understand the straightforward facts, to understand even the basic character of the physical laws in modern physics. Because stupidity became more fashionable than ever before, we may expect increasingly hopeless idiots who will be whining about the "evil Copenhagen Interpretation" in the coming decades or centuries.